Decidability A decision problem is a problem with a YES/NO answer. We have seen decision problems for - regular languages: - context free languages: [Sections 5.4 and 8.3]

Decidability A real-life decision problem (important e.g. for cryptography): “Given is a number p, is p a prime ?” How to view this problem as a language ? Which class of languages describes decidable problems ? [Section 10.5]

Decidability Are all decision problems decidable ? (I.e., is every language recursive ?) [Section 10.5]

The Halting Problem Input: a program P and an input x Answer: YES if P halts on x, otherwise (i.e. P goes into an infinite loop on x), NO. Note: the problem is typically phrased for Turing machines, i.e. P is a TM. [Sections 11.1 and 11.2]

Variants of the Halting Problem Input: a Turing machine T and an input x Answer: YES if x 2 L(T), otherwise NO. [Sections ]

Variants of the Halting Problem Input: a Turing machine T Answer: YES if T halts on the empty input, otherwise NO. [Sections ]

Variants of the Halting Problem Input: a Turing machine T, an input x, and a state q Answer: YES if the computation of T on input x reaches the state q, otherwise NO. [Sections ]

Variants of the Halting Problem Input: a Turing machine T Answer: YES if L(T) = ;, otherwise NO. [Sections ]

Variants of the Halting Problem Input: two Turing machines T 1 and T 2 Answer: YES if L(T 1 ) = L(T 2 ), otherwise NO. [Sections ]

Rice’s Theorem Rice’s Thm: Let R be a property on languages. If R holds for some but not all recursive enumerable languages, then the following problem is undecidable: Input: a Turing machine T Answer: YES if L(T) has the property R, otherwise NO. [Section 11.4]

Post’s Correspondence Problem Post’s Correspondence Problem (PCP) is a string matching problem: Input: a collection of k types of dominoes, each domino contains two strings (  i is at the top of the i-th domino,  i is at the bottom; i=1,…,k). Output: YES, if it’s possible to find a nonempty sequence of dominoes so that the string at the top matches the string on the bottom. (The domino types in the sequence can repeat.) An easy example: [Section 11.5]

Post’s Correspondence Problem Some more examples: Another example: [Section 11.5]

Post’s Correspondence Problem Thm: The Post’s Correspondence Problem is undecidable. Is PCP recursively enumerable? [Section 11.5]

Undecidable Problems for CFL’s Ambiguity of context-free grammars Input: a CFG G Output: YES if G is ambiguous, otherwise NO. Consider an instance of a PCP: (  1,  1 ), (  2,  2 ), …, (  k,  k ) Create the following CFG: S  A | B A   1 A a 1 |  2 A a 2 | … |  k A a k |  1 a 1 |  2 a 2 | … |  k a k B   1 B a 1 |  2 B a 2 | … |  k B a k |  1 a 1 |  2 a 2 | … |  k a k [Section 11.6]

Undecidable Problems for CFL’s 1.Given a CFG G, is G ambiguous ? 2.Given a CFG G, is L(G) =  * ? 3.Given two CFG’s G 1 and G 2, is L(G 1 ) = L(G 2 ) ? 4.Given two CFG’s G 1 and G 2, is L(G 1 ) Å L(G 2 ) = ; ? [Section 11.6]